2020. Т. 26. № 1. С. 9–35 2020, Vol. 26, No. 1, pp. 9–35
УДК 16 DOI: 10.21146/2074-1472-2020-26-1-9-35
Философия и логика
Philosophy and logic
Diderik Batens
Devising the set of abnormalities for a given defeasible rule
Diderik Batens
Centre for Logic and Philosophy of Science, Universiteit Gent, Blandijnberg 2, 9000 Gent, Belgium.
E-mail: Diderik.Batens@UGent.be
Abstract: Devising adaptive logics usually starts with a set of abnormalities and a deductive logic. Where the adaptive logic is ampliative, the deductive logic is the lower limit logic, the rules of which are unconditionally valid. Where the adaptive logic is corrective, the deductive logic is the upper limit logic, the rules of which are valid in case the premises do not require any abnormalities to be true. In some cases, the idea for devising an adaptive logic does not relate to a set of abnormalities, but to one or more defeasible rules, and perhaps also to one of the deductive logics. Defeasible rules are not universally valid, but are valid in
‘normal situations’ or for unproblematic parts of premise set. Where the idea is such, the set of abnormalities has to be delineated in view of the rules. The way in which this task may be tackled is by no means obvious and is the main topic studied in the present paper.
The outcome is an extremely simple and transparent recipe. It is shown that, except for very special cases, the recipe leads to an adequate result.
Keywords: adaptive logics, defeasible reasoning, defeasible rules, conditional derivation, dynamic proofs, abnormalities, falsehood, content guidance
For citation: Batens D. “Devising the set of abnormalities for a given defeasible rule”, Logicheskie Issledovaniya / Logical Investigations, 2020, Vol. 26, No. 1, pp. 9–35. DOI:
10.21146/2074-1472-2020-26-1-9-35
Envoi
This paper is dedicated to the memory of Alexander Karpenko. We got to know each other better when Alexander was responsible for three Moscow institutes in a European project ran by me on behalf of my home university, Ghent, and Salzburg, and Brussels (VUB). I still treasure a booklet with poems by Bielo Cardinal — the White Cardinal, an allusion on Alexander’s home country Belarus. I cannot read the poems, let alone understand them. Yet, at
© Batens D.
a dinner party in his home, the poet read some of them to me, and I associated them with the poems of one of my favourite writers in my home tongue, Willem Elsschot, who, apart from a fat volume of novels, left us some twenty impressive poems.
While Sasha declaimed his poems, he became for me the symbol of man reaching for what cannot be attained. That we reach anyway may, more than anything else, makes our lives meaningful. It incites and motivates us to stand by our fellow humans, to build a better world and to create beauty.
1. Aim Of This Paper
When awake, humans are in a conscious or semi-conscious state. In that state, their brain activity leads to results of many sorts: perceptions, observa- tions, goals, plans, decisions, etc. Philosophers try to explicate most of that brain activity in terms of reasoning. The bulk of this reasoning is defeasible, not deductive.
Allow me to list1 some reasoning forms that are unavoidably defeasible.
One first thinks of all kinds of inductive reasoning [Batens, 2004; Batens, 2005; Batens, 2011; Batens, Haesaert, 2003; Meheus, 2004], including induct- ive generalization as well as all predictions derived from the obtained gener- alizations. There is also abductive reasoning, with its ties to explanations of sorts [Batens, 2017; Beirlaen, Aliseda, 2014; Lycke, 2012; Meheus, 2007; Me- heus, 2011; Meheus, Batens, 2006; Meheus et al., 2002; Gauderis, Van De Putte, 2012]; but just as well approaches to explanation that do not rely on abduc- tion [Batens, 2005; Batens, Meheus, 2001; Weber, De Clercq, 2002; Weber, Van Dyck, 2001]. A very different topic is compatibility, including inconsistent com- patibility. Even finding out whether, in general, a predicative set of statements is inconsistent or not, or whether two predicative sets are incompatible with each other or not requires defeasible reasoning [Batens, Meheus, 2000; Meheus, 2003; Meheus, Provijn, 2004]. Further examples concern the logic of ques- tions [De Clercq, Verhoeven, 2004; Meheus, 2001], handling deontic conflicts [Beirlaen, Straßer, 2013a; Beirlaen, Straßer, 2013b; Goble, 2014; Meheus et al., 2010a; Meheus et al., 2010b; Straßer, 2010; Straßer et al., 2012; Van De Putte et al., in press; Van De Putte, Straßer, 2012] and many more. A whole different family are corrective adaptive logics, like the one for handling inconsistency, started in the 1980s [Batens, 1985; Batens, 1986; Batens, 1989] and having resulted in too many papers to refer to in the present context, and those hand- ling ambiguity [Batens, 2002; Vanackere, 1999a; Vanackere, 1999b; Vanackere, 2000; Vanackere, 2001].
1The interspersed references are incomplete, even with respect to adaptive logics proposed for handling the topics.
The adaptive logics programme is one of the attempts to unify all sensible and useful defeasible reasoning. It is rather easy to devise a manifold of model- theoretic, procedural, and other systems that define defeasible reasoning forms that no one could possibly unify. All those systems may prove to be interesting and even useful mathematical structures in some more or less distant future.
They may also turn out idle tea table talk. So I propose to spend a reasonable part of our efforts to defeasible reasoning forms that are known to be sensible and useful.
Adaptive logics in standard format — see Section 2. — form a unifying structure that is simple and formal. This requires some comments. The relation between the premises and the conclusion of defeasible reasoning is known to be complex. If the explication in terms of adaptive logics is right, as present insights suggest, the complexity of the consequence relation if up to Π11-complex [Batens et al., 2009; Horsten, Welch, 2007; Odintsov, Speranski, 2012; Odintsov, Speranski, 2013; Verd´ee, 2009]. Yet the ideas behind the se- mantics are transparent and unsophisticated. Moreover, there are dynamic proofs. In some cases, the proofs only stabilize at an infinite point — an un- avoidable effect of the complexity of the consequence relation. Yet the finite proof stages offer arguably a sensible estimate, in view of the information re- vealed by the stage, of the result obtained when the proof stabilizes — this is called final derivability. And indeed, proof stages are constructed by simple means. All rules are finitary — unlike for, for example, second order logic.
And which lines are IN or OUT in the any given stage of the dynamic proof is decidable. So this basically reflects the human condition: drawing conclu- sions from the available information is rather unproblematic, but we know this information to be partial and presumably misguided.
I stated that adaptive logics form a formal unifying structure. This means what it always meant: that inferences are correct in view of their form. This doesnot entail, as some simpletons actually expect, that Uniform Substitution (US) holds. US does not even work for full Classical Logic,CL.2 But a different formal criterion strictly obtains; my preferred name for it isbijective uniformity.
Technicalities aside, two inference statements Γ ` A and ∆ ` B have the same characteristic form iff each of them can be obtained from the other by systematically replacing a referring term by another referring term of the same sort — an individual constant by an individual constant, a predicate of rank r by a predicate of rank r, etc. The result is that, for example, even the propositional inference statements ¬p∧q, p∨r `r and ¬p∧p, p∨r `r donot
2The closest that comes to it is, to the best of my knowledge, still reported by Witold Pogorzelski and Tadeusz Prucnal [Pogorzelski, Prucnal, 1975]; enjoy.
have the same characteristic form because the former cannot be obtained from the latter by any suchsystematic replacement.
Given the importance of defeasible reasoning, and hence of adaptive logics in standard format as candidates for the unification, it is essential to delineate ways todevise adaptive logics. A general feature about defeasible reasoning is that it capitalizes on the fact that a certain feature or situation is normal in the sense of frequently occurring, whereas abnormal features or situations are exceptional. This leads to the idea to consider certain conclusions are justified in view of the presumed absence of abnormality. Most studied adaptive logics were obtained by first delineating the set ofabnormalities, which is characterized by a certain logical form. Thus, even if it turns out that a theory (or data set) requires∃x(P x∧Qx)∧ ∃x(P x∧ ¬Qx) to be true, one may still presume that
∃x(P x∧Rx)∧ ∃x(P x∧ ¬Rx)is false.
Next, one studies which inferences are defeasibly correct, that is cor- rect in view of the presumed falsehood of certain abnormalities. Clearly,
∃x(P x∧Rx) `CL ∀x(P x ⊃ Rx) ∨(∃x(P x∧Rx)∧ ∃x(P x∧ ¬Rx)). So if one may, reasoning systematically, consider ∃x(P x∧ ¬Rx) as false, and one knows that ∃x(P x∧Rx) is true, one may conditionally derive ∀x(P x⊃Rx).
The justification will go as follows. From the true ∃x(P x ∧ Rx) follows
∀x(P x ⊃Rx)∨(∃x(P x∧Rx)∧ ∃x(P x∧ ¬Rx)). The second disjunct of the conclusion is an abnormality, which we presume to be false and this presump- tion can be upheld. So, in the light of present insights, ∀x(P x ⊃Rx) is true.
Needless to say, this is merely an intuitive description. The matter will be phrased precisely in Section 2. and references to proofs will be given there.
So the traditional approach was to start from a set of abnormalities and next to study which defeasible inferences are correct if certain abnormalities may be presumed to be false. As becomes clear in the next section, once we know what the abnormalities are, the relevant adaptive logic is easily defined.
Adaptive logics consider abnormalities as false in ‘normal’ situations; as false until and unless proven otherwise.
Often, however, in devising an adaptive logic, one does not know from the beginning which are the abnormalities. Rather, one knows that the reasoning step A/B3 is correct when ‘nothing is wrong’, when the situation is normal.
Here “normal” points to a further unknown situation, the situation in which the rule A/Bis valid.
Concrete examples follow in subsequent sections, but the problem is to collect general insights on the relation between the abnormalities and such a defeasible rule. Does the rule determine the set of abnormalities? Do several
3Rules are phrased in metalinguistic terms. So I use meta-metalinguistic variables for formulas to describe a rule.
sets of abnormalities make the rule valid as a defeasible rule? If so, what are the effects of different choices?
2. Preliminaries
Many introductions to adaptive logics are available — the most recent one is always the best [Batens, 2015]. So I shall be very brief here. Moreover, the reader may skip this section and look up things in it as she or he needs them to understand subsequent sections.
An adaptive logic,AL, in standard format is a triple:
(i) a lower limit logic LLL: a logic that has static proofs and has a nice semantics;4
(ii) a set of abnormalities Ω: a decidable set of formulas characterized by a (possibly restricted) logical form F; or a union of such sets;5
(iii) anadaptive strategy: Reliability or Minimal Abnormality.6
The upper limit logic ULL is obtained by extending the lower limit lo- gic LLL with an axiom stating that all abnormalities cause triviality. Where a premise set Γ does not require any abnormalities to be true, the AL- consequences of Γ provably coincide with its ULL-consequences. One of the effects is that the inconsistency-adaptive consequences of a consistent premise set coincide with the set’s CL-consequences.
In a ‘Dab-formula’ Dab(∆),∆ is a finite subset ofΩ and Dab(∆) denotes theclassical disjunction of the members of∆. So classical disjunction needs to be present in the language or has to be added.7
Dab(∆) is a minimal Dab-consequence of Γ iff Γ `LLL Dab(∆) whereas Γ 0LLL Dab(∆0) for any ∆0 ⊂ ∆. A choice set of Σ = {∆1,∆2, . . .} is a set that contains an element out of each member of Σ. A minimal choice set of Σ is a choice set of Σ of which no proper subset is a choice set of Σ.
WhereDab(∆1),Dab(∆2), . . . are the minimalDab-consequences ofΓ,U(Γ) =
∆1∪∆2∪. . . andΦ(Γ)is the set of minimal choice sets of Σ ={∆1,∆2, . . .}.
4Read this as a compact Tarski logic with a characteristic semantics. The idea of a nice semantics [Verd´ee, Batens, 2016] is more sophisticated than that of a characteristic semantics and is fascinating in view of its implications for embedding. Unfortunately, explaining it here would require too long a digression.
5WhereFais the set of atomic formulas (those containing no logical symbols other than
=),{A∧ ¬A|A∈ Fa}is an example of a restricted logical form.
6There are the most important strategies.
7As Sergei Odintsov and Stanislav Speranski first pointed out [Odintsov, Speranski, 2013], an alternative is to phrase adaptive logics in multiple conclusion terms.
Definition 1. A LLL-model M ofΓ isreliable iffAb(M)⊆U(Γ).
Definition 2. ΓALr A iff Ais verified by all reliable models of Γ.
Definition 3. A LLL-model M of Γ is minimally abnormal iff there is no LLL-model M0 of Γsuch thatAb(M0)⊂Ab(M).
Definition 4. Γ ALm A iff A is verified by all minimally abnormal models of Γ.
It can be shown that a LLL-model M of Γ is minimally abnormal iff Ab(M)∈Φ(Γ).
Although I started with their semantics, adaptive logics were discovered by reflecting on dynamic proofs — the theorizing on dynamic proof theories came much later [Batens, 2009]. An annotated AL-proof consists of lines that have four elements: a line number, a formula, a justification (at most referring to preceding lines) and a condition. Where
A ∆
abbreviates that A occurs in the proof as the formula of a line that has ∆as its condition, the (generic) inference rules are —∨ˇ is a classical disjunction:
PREM If A∈Γ: . . . .
A ∅
RU If A1, . . . , An`LLLB: A1 ∆1
. . . . An ∆n
B ∆1∪. . .∪∆n
RC If A1, . . . , An`LLLB∨Dab(Θ)ˇ A1 ∆1
. . . . An ∆n
B ∆1∪. . .∪∆n∪Θ
A proof stage is a list of lines obtained by applications of the generic rules PREM, RU and RC. Let the empty list be stage 0. Where s is a stage, s0 is an extension of s iff all lines that occur ins occur in the same order in s0. A (dynamic) proof is a chain of stages. That A is derivable on the condition ∆ may be interpreted as: it follows from the premise set that A or one of the members of∆is true. Because the members of∆, which are abnormalities, are presumed to be false,Ais considered as derived, unless and until it shows that
the presumption cannot be upheld. The precise meaning of “cannot be upheld”
depends on the strategy, which determines the marking definition (see below) and hence determines which lines are marked at a stage. If a line is marked at a stage, its formula is considered as not derived at that stage.
Dab(∆)is aminimalDab-formula at stagesof anAL-proof iffDab(∆)was derived ats on the condition∅whereas for no∆0 ⊂∆wasDab(∆0) derived on the condition ∅. Where Dab(∆1), . . . ,Dab(∆n)are the minimal Dab-formulas at stage s of a proof from Γ, Us(Γ) = ∆1 ∪. . .∪∆n and Φs(Γ) is the set of minimal choice sets of{∆1, . . . ,∆n}.
Definition 5. Marking for Reliability: Linel is marked at stages iff, where∆ is its condition,∆∩Us(Γ)6=∅.
Definition 6. Marking for Minimal Abnormality: Line lis marked at stage s iff, where A is derived on the condition ∆on line l, (i) there is noϕ ∈Φs(Γ) such that ϕ∩∆ =∅, or (ii) for some ϕ∈Φs(Γ), there is no line on whichAis derived on a condition Θfor whichϕ∩Θ =∅.
Let me rephrase this: whereAis derived on the condition∆on linel, linel is unmarked at stage s iff (i) there is a ϕ ∈ Φs(Γ) for which ϕ∩∆ = ∅ and (ii) for everyϕ∈Φs(Γ), there is a line at which A is derived on a condition Θ for which ϕ∩Θ =∅.
A formulaAisderived at stage sfromΓiff it is the formula of a line that is unmarked ins. Marks may come and go as the proof proceeds. Yet there also is a stable notion of derivability, called final derivability.
Definition 7. A is finally derived from Γ on line l of stage s iff (i) A is the second element of line l, (ii) line l is not marked at stage s, and (iii) every extension of the stage in which line l is marked may be further extended in such a way that line lis unmarked.
Definition 8. Γ `AL A (A is finally AL-derivable from Γ) iff A is finally derived on a line of a proof stage from Γ.
There are three comments in conclusion of the preliminaries. First, ad- aptive logics are not competitors of deductive logics, but means to arrive at formal characterizations of methods. Next, one typically needs adaptive lo- gics (and, more generally, defeasible reasoning) when a positive test is absent.
Consider any of the examples mentioned before. At the predicative level, the consequence set of the adaptive logics is not semi-recursive. The final comment is that adaptive logics have an impressive metatheory which required the de- velopment of novel proof methods. The metatheory includes Soundness and Completeness proofs, but also the proofs of many features that are entirely
foreign to deductive logics. I refer to [Batens, 2007] for the metatheory and to [Batens, 2015] for a revised formulation of the theorems, often leaving the straightforward reformulation of the proofs to the reader.
3. The Problem
Consider a defeasible ruleA1, . . . ,An/Bthat we consider as valid innormal situations. In rather exceptional cases we have no precise idea of the lower limit logicLLL, but let us neglect that problem and suppose that the strategy as well as LLL are given.8 The task is to find the form of a formula C that may serve as the abnormality for the rule, viz. such thatA1, . . . ,An`LLLB∨C.ˇ Once this C is found, the corresponding conditional rule CR will be: “from A1, . . . ,An on the condition∆to derive Bon the condition∆∪ {C}”.
The reader may wonder whether a single abnormalityCis introduced rather than a Dab-formula, as was suggested by the way the generic rule RC was phrased in the previous section. This is an interesting point. Let us leave open whetherCwill be the form of the abnormalities or whetherCmay indeed be itself a disjunction of abnormalities. Let us also leave open whether the problem is to find a uniqueC or several — the latter case refers to the second alternative in the description of the set of abnormalitiesΩ: “or a union of such sets”. The first sentence of the present paragraph moreover reminds us of an important matter. We want to delineate Ω in function of the defeasible rule A1, . . . ,An/B. Yet, we are after an adaptive logic in standard format. In other words, the generic rule RC will by no means be restricted to the defeasible rule A1, . . . ,An/B. The generic rule RC will solely depend on LLL and Ω as is obvious from Section 2. We shall see that this consideration will play an important role in subsequent pages.
Consider some examples of defeasible rules in the domain of inductive gen- eralization. There are many adaptive logics in that domain. Each of them characterizes a way to defeasibly infer generalizations. A generalization is a for- mula∀xA(x)in whichA(x)is a truth function of literals in which no individual constants occur.9 One of the logics allows one to introduce generalizations as Popperian hypotheses, the defeasible rule then becoming −/∀xA(x) — given whatever, one may conditionally introduce a generalization. Other such logics require an instance and hence need a defeasible rule A(α)/∀xA(x), in which α may be any individual constant. Still, other logics require a ‘positive instance’
8When we are after an ampliative adaptive logic, LLL will be the deductive logic we consider suitable in the given context. For many this will beCLwhen the context concerns empirical or classical mathematical theories.
9The precise formulation was published elsewhere [Batens, 2011], but is not terribly im- portant in the present context.
as inB(α)∧C(α)/∀x(B(x)⊃C(x)). In all of these Aand C are disjunctions of one or more literals andB is a conjunction of literals — conjunctions of two generalizations are derived by RU from generalizations derived by RC. So, for each such defeasible rule, the task is to pinpoint an abnormality, which then will determine the set of abnormalitiesΩ for that logic.
The sets of abnormalities for those inductive generalization rules were de- lineated a long time ago by tinkering. This was not difficult and they agree nicely with the recipe that will be presented in the present paper. This is a good reason to consider a different type of adaptive logics.
It is desirable to refer to a case where the matter becomes slightly more difficult as well as slightly more interesting. While working on adaptive set theories [Batens, 2019], I came about a case that I never met before. That we are dealing with a corrective adaptive logic is a difference with the logics from the previous paragraphs. Yet, something is more important. The lower limit logic of the set theories is the paraconsistent CLuNs, which is specified be- low, and the strategy is Minimal Abnormality. The well-studied inconsistency- adaptive logic CLuNsm is obtained by specifying the set of abnormalities as {Q(A∧ ¬A) | A ∈ Fa}, in which Fa is the set of (open and closed) atomic formulas andQ(A)is(A)preceded by a quantifier over every formula free inA.
I give this set a specific name for future reference. It turns out that certain premise sets require a different adaptive logic, one that has a more embracing set of abnormalities and hence assigns a richer consequence set to the premise sets.
While adaptive logics were originally devised as ways to formally charac- terize methods, it turned out that they may also be profitably invoked to char- acterize complex theories — viz. theories that are not semi-recursive. Partly relying on work by others, I made attempts to devise adaptive theories for Peano Arithmetic and for Frege’s notion of a set. It is the latter that led to the case I now shall outline. I’ll just mention some ideas, as the paper will soon be available in print. However, there are some details I have to report explicitly in order to clarify the problem. Readers who are in a hurry may skip to the beginning of Section 4. and return here later if they get interested in the significant example.
As Frege’s notion of a set makes inconsistent sets unavoidable, the lower limit logic of the adaptive logic needs to be paraconsistent. For reasons not discussed here, I choose the (very popular) paraconsistent logic (which I prefer to call)CLuNs[Batens, Clercq, 2004]. Apart from negation, all logical symbols
are exactly as inCLand RoI (Replacement of Identicals) holds unrestrictedly.10 The negation ¬ is strictly paraconsistent11 and reduces complex negations to simpler ones in the usual way: ¬¬A ≡ A, ¬(A∧B) ≡ (¬A∨ ¬B), . . . and
¬∃xA≡ ∀x¬A.
The set theory obtained by CLuNsfrom (a version of) the Fregean axiom schema Abs and axiom Ext will be called PFS (paraconsistent Fregean set theory).12 Obviously, one wants to move from the paraconsistent theory to an adaptive one, call it AFS. While one has unavoidably to allow for some inconsistent sets — sets of which some members are also non-members — one wants that sets are only inconsistent when this is unavoidable, and one wants even inconsistent sets to behave as consistently as possible. For example, one wants∅to be consistent and, while the Russell setRis unavoidably inconsistent, one wants∅∈/R in view of∅∈ ∅/ and one does not want ∅ ∈R.
Just like the language of most mathematical theories, the language of set theory is extremely simple. Apart from the logical symbols and the variables of the standard predicative language, it has one binary predicate ∈ and of- ten abstracting terms of the form {α | A(α)}. Where the underlying logic is CLuNs, some formulas of this language express triviality,13 for example
∀x∀y(x=y∧x6=y∧x∈y∧x /∈y), which I shall abbreviate as⊥.14 Literally every formulaof the set theoretical language isCLuNs-derivable from this (as well as from some other formulas).15 Given that material implication is present with all itsCL-properties, classical negation can be defined: ¬Aˇ =df A⊃ ⊥.
The presence of classical negation has the unexpected consequence that the Abs axiom requires the existence ofR∗ =df {x|¬ˇx∈x}. While inconsistency results, R∗ ∈ R∗∧R∗ ∈/ R∗, it is provable that R∗ ∈ R∗∧¬Rˇ ∗ ∈ R∗ is not derivable and that the inconsistency-adaptive theory is non-trivial, just like the paraconsistent theory. Yet, the fact that R∗ ∈ R∗ is a theorem of the
10RoI:a=b⊃(A≡Aa/b) in whichAa/b is the result of replacing inAan occurrence of abybor vice versa. In some paraconsistent logics, RoI does not hold within the scope of a negation.
11A negation¬is paraconsistent iffA,¬A`B does not hold for allAandB; it is strictly paraconsistent iff there is noAsuch thatA,¬A`B holds for allB.
12WithinCLuNs there are three different implications that coincide in CL: A ⊃ B is detachable but not contraposable,A=B =df ¬A∨B is not detachable but contraposable, A→B=df (A⊃B)∧(¬B⊃ ¬A)is both detachable and contraposable; similarly, there are 16 different equivalences that coincide inCL. So choices have to be made as one moves from Frege’s trivial theory to the provably non-trivialCLuNs-theoryPFS.
13Several other paraconsistent logics have the same property.
14The abbreviationst16=t2=df ¬t1=t2andt1∈/t2=df ¬t1∈t2 occur for readability.
15The formula does not express triviality in some extensions of the language of set theory.
So it is a remarkable case of expressing local triviality, a feature that also occurs in other mathematical theories — I shall soon publish a brief study of the remarkable phenomenon and its epistemic potential.
paraconsistent theory, and hence also of the adaptive one, reveals a perhaps unpleasant but interesting phenomenon: R∗ has members that do not fulfil the touchstone of R∗. Indeed, R∗ ∈R∗ is a theorem of the paraconsistent theory, but R∗ does not fulfil the touchstone, which is ¬Rˇ ∗ ∈R∗; and it cannot fulfil the touchstone — the theory is non-trivial. I shall say that R∗ is clean iff
∀y(y ∈ {x | A(x)} ↔ A(y)), in which A ↔ B =df (A ≡ B)∧(¬B ≡ ¬A).
It turns out that both ∀y(A(y) ⊃ y ∈ {x |A(x)}) and ∀y(¬A(y) ⊃y /∈ {x | A(x)})can be required to hold, but not their converses, precisely because some sets, for example R∗ are unavoidably unclean. The converses have to read
∀y(y ∈ {x|A(x)}=A(y))and ∀y(y /∈ {x|A(x)}=¬A(y))— remember that
=is not detachable.16
This situation reveals a problem that requires a solution. Indeed, R =df {x | x /∈ x} is inconsistent but there is no need for it to be unclean. Let Ω1 =df {Q(A∧ ¬A) | A ∈ Fa}, in which Fa is the set of (open and closed) atomic formulas and Q(A) is (A) preceded by a quantifier over every formula free in A. Consider a PFS-model M that is minimally abnormal with respect to Ω1. Obviously, the domainD ofM is uncountable17 whence some elements ofDhave no name — are not named by an abstracting term. It turns out that some sets are clean in some PFS-models that are minimally abnormal with respect to Ω1, but are unclean in other PFS-models that are also minimally abnormal with respect to Ω1. A typical example is precisely R. Consider a minimally abnormal PFS-model M1 in which R is clean and consider an elemento∈Dthat stands in the∈-relation toRand not also in the ∈-relation/ to R — technically this will be expressed for example by ho, v(R)i ∈vT(∈).18 Next consider a model M2 that is exactly likeM1 except in that ois not only a member but also a non-member ofR. So, inM2, the setR is unclean as well as inconsistent. Yet, given that no individual constant of the language refers to o, the inconsistency can only be stated as ∃x(x ∈ R∧x /∈ R). But this formula is also verified byM1, because all PFS-models verify R∈R∧R /∈R.
So R is clean in M1 and is unclean in M2, but both are minimally abnormal and actually Ab(M1) = Ab(M2). This is not as we want it. The axioms do not require R to be unclean. They do not even require that R is a member
16It is not really essential to this paper, but Abs comes to the conjunction of the four implications mentioned in the text, two detachable ones and two non-detachable ones.
17Many uncountableZF-hierarchies can be defined in exactly the same way inPFSand if their members were countable inPFS, then they would be inconsistent. It can be argued that they are consistent in minimally abnormal models ofPFSif they are consistent inZF.
18In this semantic style, the extension of a predicateπ of rank n is a triplehΣ1,Σ2,Σ3i withΣ1,Σ2,Σ3⊇DnandΣ1∪Σ2∪Σ3=Dn. Next, for convenience, the assignment function v(π)is seen as three functions: vT(π) = Σ1,vB(π) = Σ2 andvF(π) = Σ3.
of a minimal set of sets one of which is bound to be unclean. So R should be clean.19
What does this come to? We want the non-detachable∀y(y ∈ {x|A(x)}= A(y)) to have the strength of the detachable ∀y(y ∈ {x | A(x)} ⊃ A(y)) as much as possible; and similarly for upgrading ∀y(y /∈ {x | A(x)} = ¬A(y)) to ∀y(y /∈ {x | A(x)} ⊃ ¬A(y)). Obviously, we do not want to interpret all expressions A = B as much as possible as A ⊃ B. We only want instances of schema Abs to be as much as possible detachable in all directions. Abs was intended originally as detachable in all directions. This cannot be realized completely because Frege’s notion of the extension of a predicate turned out inconsistent. Nevertheless, the original intention can be realized as much as possible. But this cannot be done by minimizing inconsistencies: remember that M1 and M2 verify the same members of Ω1. We have a case where, for ordered pairs hA, Bi of certain forms, we want to derive A ⊃ B from A = B on a certain condition. The task is to find the condition.
4. Solving the Problem
Let us concentrate first on rules with one local premise. The task then is, starting from the defeasible rule A/B, to find a condition or conditions C such that the three following hold:
A`LLL B∨Cˇ and A0LLL B and A0LLL C. (1) IfA`LLL B, thenA/BisLLL-valid and not a defeasible rule. IfA`LLLC, then the conditionCcauses the ruleA/Bto have no sensible applications. Remember indeed that, as stated in Section 3., the idea is to obtain the following particular conditional rule: “from A on the condition ∆ to derive B on the condition
∆∪ {C}”. If the defeasible ruleA/Bis to have sensible applications, there must be a premise set Γ and a (empty or non-empty) ∆ ⊂ Ω such that (i) A is finally derivable fromΓ on the condition∆and (ii) Bis finally derivable from Γ on the condition ∆∪ {C}. In view of (i), Γ `LLL A∨Dab(∆)ˇ and there is a ϕ ∈ Φ(Γ) such that ∆∩ϕ = ∅. In view of (ii), Γ `LLL B∨Dab(∆ˇ ∪ {C}) and there is aϕ∈Φ(Γ)such that (∆∪ {C})∩ϕ=∅. The latter is impossible because, if A `LLL C, then Γ `LLL A∨Dab(∆)ˇ is a sufficient condition for Γ`LLLC∨Dab(∆), which isˇ Γ`LLLDab(∆∪ {C}). WhetherDab(∆∪ {C})is a minimal Dab-consequence of Γ or not, everyϕ∈Φ(Γ) contains at least one
19The difference betweenM1 and M2 cannot be expressed in the language by a formula stating a contradiction, whether plain or quantified. Does this mean that the difference betweenM1andM2cannot be expressed? By no means. We just need different abnormalities.
member of Dab(∆∪ {C}). But then the line at which B is derived byA/B is always marked. This ends the justification of the requirements in (1).20
Let us move to a concrete case, viz. the defeasible rule
A=B/A⊃B . (2)
We are looking for one or more conditions such that the three derivability statements in (1) are fulfilled. I shall first consider the problem in the context of the lower limit logicCLuNs, with the classical negation¬ˇ present or added, and the Minimal Abnormality strategy, but neglecting for the moment that the problem arose in connection with the set theory AFS.
In the previous section, (2) was considered in a situation in which the ab- normalities were contradictions, as is usual for inconsistency-adaptive logics.
Some people will keep repeating that abnormalities of the formA∧ ¬Ajustify the defeasible rule (2). Indeed,
A=B `CLuNs(A⊃B)ˇ∨(A∧ ¬A)
holds. Or, even more explicitly in view ofCLuNs-equivalences,
¬A∨B`CLuNs(ˇ¬A∨B)ˇ∨(A∧ ¬A).
However, and as already explained in Section 3., this is not the point.21 The point is that, for specific ordered pairs hA, Bi, we want (2) to be applied even if A∧ ¬Ais true.22
We are in search of conditions C that fulfill (1) and, by their forms, de- termine a setΩthat, together with the lower limit logic CLuNsand the Min- imal Abnormality strategy, defines an adequate adaptive logic. We need a C such thatA`CLuNsB∨C. Given thatˇ CLuNshas a nice semantics and given Soundness and Completeness,A`CLuNsB∨Cˇ is equivalent toA∧ˇˇ¬B`CLuNsC.
The strongest such C is obviously A∧ˇˇ¬B itself and every such C is aCLuNs- consequence of A∧ˇˇ¬B.
Let us apply this at once to the defeasible rule (2). The strongest condition C is (A =B)∧¬(Aˇ ⊃B), which is CLuNs-equivalent to (A∧ ¬A∧¬Bˇ ) — as the conjunction is classical in CLuNs, there is no need to write ∧. Soˇ the defeasible rule phrased with its strongest condition reads: “from A on the condition ∆to derive B on the condition ∆∪ {A∧ ¬A∧¬B}”. Actually, theˇ
20The justification considers only Minimal Abnormality. Where Reliability is the strategy, the justification is much simpler and left to the reader.
21Moreover and concerningAFS, every unclean set is unavoidably inconsistent: ift∈ {x| A(x)}but¬A(t), thent /∈ {x|A(x)}. However, this too is not the point.
22InAFSwe want all sets to be as clean as possible, even inconsistent sets.
logic CLuNsrequires23 that the set of abnormalities contains all formulas of the formA∧ ¬A∧¬Bˇ in whichA is an atomic formula and B is a literal.24
Next, consider weaker conditions C for the defeasible rule (2). Let us have a systematic look at ‘parts’ ofA∧ ¬A∧¬Bˇ . The idea is not to find aC that in itself gives us all we want, but to find conditionsCthat are acceptable, possibly in the presence of other conditions. I neglect the fact that the local premise may come on a condition itself; by now, the reader will have understood the resulting complication. To the left is the CLuNs-inference, to the right the effect on a dynamic proof.
A=B (A⊃B)∨C
A=B A⊃B {C}
Let us consider the possibilities systematically.
(i) We know already that the strongestC isA∧ ¬A∧¬B.ˇ
(ii) That C is A∧ ¬A is all right provided one also wants all conditional inferences that then are correct in view of the standard format, specifically RC. An example is the effect ofA⊃B,¬B `CLuNs¬A∨(B∧ ¬B): from A ⊃B and ¬B to derive ¬A on the condition{B ∧ ¬B}. So what this comes to is that the choiceA∧ ¬Ais all right in case one agrees that ¬A has actually the force of ¬Aˇ whenever A∧ ¬A can be taken to be false.
(iii) That C is A∧¬Bˇ is not acceptable. Indeed, this condition is simply the classical negation of the conclusion of the defeasible rule. Once ¬ˇ is added to CLuNs, (A ⊃ B)∨(A∧¬B)ˇ is a CLuNs-theorem. So if A∧¬Bˇ is an abnormality, possibly withA restricted to atomic formulas and B to literals, then A ⊃ B is derivable on the condition {A∧¬B}ˇ from any premise set. Unlike what the reader might expect, this would not cause premise sets to have trivial consequence sets; most conditional lines would be marked. Yet, there is no sensible idea behind this choice of an abnormality and the choice does not seem to lead to anything sensible.
Nevertheless, I shall return to this choice below.
23The requirement is related to the avoidance of so-called flip-flop adaptive logics, which are only desirable for specific applications [Batens, 2007]. The point need not further concern us here.
24The set of literals is the set of non-equivalent formulas in which occurs an atomic formula preceded by at most unary connectives. Where two negations,¬ and¬, are present in theˇ language ofCLuNs, the notion of a literal is a trifle more sophisticated than inCL. While this set is{A,¬A}(Aa sentential letter) in CL, it is{A,¬A,¬A,ˇ ¬¬A,ˇ ¬ˇˇ¬A,¬ˇˇ¬¬A}(Aa sentential letter) inCLuNs.
(iv) Somewhat unexpectedly, it seems all right at first sight to choose¬A∧¬Bˇ as C. Indeed, when one concentrates on the defeasible rule we are con- sidering here, the choice seems unobjectionable, both in case ¬A∧¬Bˇ is true and in case it is false. In the latter case, for example, we obtain: if
¬A is false, the premise warrants thatB is true; if¬Bˇ is false, thenB is also true. AsBis true, so isA⊃B. And yet this choice has consequences we do not want. As¬A`CLuNsB∨(¬A∧¬Bˇ ), the choice would justify that one would derive an arbitraryB from¬Aon the sole condition that
¬A∧¬Bˇ can be taken to be false. As in the previous case, triviality would not result25but there is no idea behind this way of proceeding and nothing sensible is expected to result.
(v) To choose A or ¬A or B as C is obviously unacceptable. That literals would be abnormalities, would result in all kinds of turmoil, but in nothing sensible.
(vi) I promised to return to (iii). Choosing(A=B)∧(A∧¬B)ˇ asCprevents one to introduce detachable implications from the blue. Moreover, these abnormalities nicely express that the premise is true and the desired con- clusion false. However, nothing new is arrived at along this road. The chosen abnormality is CLuNs-equivalent toA∧ ¬A∧¬B, which is theˇ abnormality considered in (i).
No other choices are worth commenting upon. Yet it is still interesting to consider combinations, viz. that formulas of different forms are counted as abnormalities, for example A∧ ¬A and A∧ ¬A∧¬B. Neglecting some com-ˇ plications, a line is unmarked and its formula is not considered as derived iff its condition can be considered to be false. Suppose that A∧ ¬A cannot be considered as false. Then it is nevertheless possible that A∧ ¬A∧¬Bˇ can be considered as false: if A and ¬A are both true, but ¬Bˇ is false, then the conjunction of the three formulas is false.26 So allowing for abnormalities of both forms has the following effect — I keep restricting attention to crucial in- sights. On the one hand, including formulasA∧ ¬A in the set of abnormalities has the effect that a lot of further conditional inferences become valid, as was explained in (ii). On the other hand, even ifA⊃B cannot be seen as derived on the condition A∧ ¬A because this condition cannot be considered as false,
25Adaptive logics in standard format have the Strong Reassurance property (also called Stopperedness or Smoothness): if a premise set has LLL-models, then it has minimally abnormal models. Proofs were published long ago [Batens, 2000; Batens, 2007].
26Spelling the matter out in a precise way for Reliability and (especially) for Minimal Abnormality is much more complicated, but the crucial insight is the one stated in the text.
it is possible thatA⊃B can be seen as derived on the conditionA∧ ¬A∧¬Bˇ because¬Bˇ can still be considered as false.
The matter seems clarified, but there are still two little problems. I comment on them in order to illustrate the complications involved in the systematisation of defeasible reasoning. The easier problem is this: A∧ ¬A∧¬Aˇ obviously has the form of A∧ ¬A∧¬Bˇ and is always false.27 That looks frightening.
As the line will never be marked, such an adaptive logic seems to extend the lower limit logic with non-defeasible steps. However, this is a pseudo-problem.
Whenever A`CLuNsB∨C and C∈Ω is logically false, then A`CLuNsB. So logically impossible abnormalities are harmless; they are obviously also useless.
The second problem is more interesting: if formulas of the formA∧¬A∧¬Bˇ are abnormalities, does it then even make a difference whether formulas of the formA∧ ¬Aare also abnormalities? While¬A0CLuNs¬Aˇ ∨(A∧ ¬A∧¬B),ˇ it holds that ¬A,¬Bˇ 0CLuNs ¬Aˇ ∨(A∧ ¬A∧¬Bˇ ). So, if any formula of the form¬Bˇ is derivable, even if only conditionally, an unexpected effect seems to follow. Let me show this by presenting a little proof.
... ... ... ...
51 ¬sˇ . . . ∆
52 ¬p PREM ∅
53 ¬pˇ 51, 52; RC ∆∪ {p∧ ¬p∧¬s}ˇ
Supposing that line 51 is unmarked, line 53 will be unmarked just in case p∧ ¬p can be held to be false. If a formula of the form¬Bˇ is derivable, even conditionally, from the premises, then abnormalities of the form A∧ ¬A are redundant.
The matter becomes less surprising if one realizes that conditional transition from ¬p to ¬pˇ may be realized in a way that seems unobjectionable. Recall thatA=B abbreviates¬A∨B.
... ... ... ...
51 ¬sˇ . . . ∆
52 ¬p PREM ∅
53 p=s 52; RU ∅
54 p⊃s 53; RC {p∧ ¬p∧¬s}ˇ
55 ¬pˇ 51, 54; RU ∆∪ {p∧ ¬p∧¬s}ˇ
What happens here is that we apply the defeasible rule (2) at line 54 and next apply Modus Tollens — this is correct as ⊃ is detachable and ¬ˇ is classical negation.
27Many will not care about the detail, but it is more correct to say thatA∧ ¬A∧¬Aˇ has no non-trivial models — in some semantic styles no models, in others only the trivial model.
If a bottom constant ⊥ is present in the applied version of CLuNs, the matter is even easier.
1 ¬p PREM ∅
2 ¬pˇ 1; RC {p∧ ¬p∧¬⊥}ˇ
Here `CLuNs¬⊥ˇ whencep∧ ¬p∧¬⊥ˇ isCLuNs-equivalent to p∧ ¬p.
After discussing (2) independently of the set theoretical context, let us now return to the problem in AFS. Two insights are important. (i) If, reasoning systematically, it is possible to consider A∧ ¬A as false, then it is possible to consider A∧ ¬A∧¬Bˇ as false, but not conversely. So, given a premise set Γ, the more complex condition allows for final consequences of Γ that are not final consequences if all abnormalities have the form A∧ ¬A. (ii) As noted in Section 3., the aim is not to upgrade expressionsA=B as much as possible to A⊃B, but to do so only for specific formulas that are implicative parts of Abs, viz. ∀y(y∈ {x|A(x)}=A(y))and ∀y(y /∈ {x|A(x)}=¬A(y)). Consider the formulas
Q(t∈ {x|A(x)} ∧¬A(t))ˇ and Q(t /∈ {x|A(x)} ∧¬¬A(t))ˇ (3) in whichtis a set theoretical term and, if it is a variable,Qis a quantifier over that variable. In view of Abs, the left formula PFS-entails Q(t /∈ {x|A(x)} ∧ t∈ {x|A(x)}∧ˇ¬A(t))and the right formulaPFS-entailsQ(t∈ {x|A(x)}∧t /∈ {x|A(x)} ∧¬¬A(t)). So a bit of calculation shows that formulas of the formsˇ in (3) may be safely taken as abnormalities. Proceeding thus, we obtain an adaptive logic that minimizes inconsistencies in view of abnormalities of the formA∧ ¬A and minimizes uncleanness in view of the abnormalities from (3).
The conclusion then is as follows. First, unless there are very special reasons to refrain from applying the defeasible rule in its full generality, the recipe leads to the following schema, considering multiple local premises (but still restricting to the lower limit CLuNs).
A1 ∆1
... ... An ∆n
B ∆1∪. . .∪∆n∪ {A1∧. . .∧An∧ ∗B}
(4)
Where ∗¬Bˇ = B and ∗B = ˇ¬B in case the first symbol in B is not ¬. Callˇ A1∧. . .∧An∧ ∗Bthetypical abnormality for the defeasible ruleA1, . . . ,An/B.
There is no reason to prefer abnormalities obtained by dropping one or more conjuncts ofA1∧. . .∧An∧ ∗Bbecause if the shorter formula can be considered as false, then so can the longer one. When one wants to introduce several
defeasible rules, each of them may be given its typical abnormality. Two points of attention then are: (i) one should check which rules need to hold in their formal generality and which should be restricted to specific formulas — the AFScase illustrates this candidly — and (ii) one should study the effect of the typical abnormality of a rule on other rules in view of the logical and conceptual context — see the example discussed above.
The occurrence of the classical negation ¬ˇ in the typical abnormality may look like causing trouble in paraconsistent contexts, especially for dialetheists.
The matter will be discussed in Section 5.
When the lower limit logic is not CLuNs, the typical abnormality is easily adjusted. Two adjustments may be required. (i) Sometimes restrictions on the subformulas of abnormalities need to be modified or removed. Thus the restriction A ∈ Fa needs sometimes to be replaced, for example by A ∈ F, in which F is the set of (open and closed) formulas of standard predicative languages. (ii) Sometimes logical symbols are classical within the considered lower limit logic, whence we do not need ¬ˇ and ∗ — I actually applied this already within the present paper. I do not enter this any further as the matter is mainly technical.
Let us consider some typical abnormalities for other rules. I mentioned three defeasible rules of inductive generalization. In each of themA,B and C are disjunctions of literals and there are some further restrictions. The typical abnormalities for each of the rules can be read off below in somewhat simplified form — as the lower limit is CL, the standard negation is classical:
–
∀xA(x) {∃x¬A(x)}
∃xA(x) ∆
∀xA(x) ∆∪ {∃xA(x)∧ ∃x¬A(x)}
∃x(B(x)∧C(x)) ∆
∀x(B(x)⊃C(x)) ∆∪ {∃x(B(x)∧C(x))∧ ∃x¬(B(x)∧ ¬C(x))}
An inconsistency can be seen as a negation glut: that vM(A) = 1 justifies vM(¬A) = 0on the CL-semantics, but actually vM(¬A) = 1. A negation gap is where vM(A) = 0, which justifies vM(¬A) = 1 on the CL-semantics, and nevertheless vM(¬A) = 0. Along this line, one may consider gluts and gaps for every logical symbol of the standard predicative language. Adaptive logics minimizing gaps and gluts were studied [Batens, 2016]. Suppose that 99Kis a glutty implication, whereas⊃is the classical implication. A glutty implication is obviously not detachable, as there are models in which vM(A 99KB) = 1 = vM(A)andvM(B) = 0. This brings us to something very close to (2). In order to minimize implication gluts, we want the following rule and abnormality:
A99KB ∆
A⊃B ∆∪ {(A99KB)∧A∧ ¬B}
If glutty implications are combined with glutty negations (and possibly more oddities), the negation needs to be replaced by ¬. All glutty and gappy logicalˇ symbols may be given defeasible rules to minimize them and the insight gained in this section will provide the rules with typical abnormalities.
5. A Puzzle In Inconsistency-Adaptive Logics
Classical negation occurs in the typical abnormalities from the previous section. Some will see this as problematic in paraconsistent contexts. Of course, if classical negation is not definable in a paraconsistent logic, one may add it, possibly forbidding its occurrence in premises and conclusion. Yet especially dialetheists will have objections to such move as they consider classical negation as a tonk-like operator. This conclusion is related to the dialetheist view that all true knowledge should form a single body, phrased within a single language and organized by The True Logic and that this body is necessarily inconsistent in view of the Liar paradox, paradoxes of set theory, etc. I shall not discuss the dialetheist position here, but rather argue that, for two reasons, the typical abnormalities do not lead to a situation that is at odds with dialetheism.
The first reason is that, due to the structural properties and functioning of negation, the typical abnormalities do not require that classical negation ever occurs either in them or elsewhere in a proof. First of all, look at two basic defeasible rules for negation:28
¬A ∆
¬Aˇ ∆∪ {A∧ ¬A}
A ∆
¬¬Aˇ ∆∪ {A∧ ¬A} (5) However, once the adaptive logic is characterized in terms of the Standard Format, these rules need not be mentioned. Applications of the generic condi- tional rule RC may be phrased completely in the standard language, without ever writing a classical negation. Here are two examples, an application of Disjunctive Syllogism and one of Modus Tollens.
A∨B ∆
¬A Θ
B ∆∪Θ∪ {A∧ ¬A}
A⊃B ∆
¬B Θ
¬A ∆∪Θ∪ {B∧ ¬B}
The classical negation in (5) signifies that, provided the abnormality in- troduced by the condition can be held to be false, A, respectively ¬A, can be considered as consistently false; spelled out,¬Aˇ signifies thatA is consistently
28Sometimes theAin the abnormality is restricted, for example to atomic formulas, as is required whenCLuNsis the lower limit logic. Sometimes several defeasible rules are required as inAFS.
false and ¬¬Aˇ that A is consistently true. In the application of Disjunctive Syllogism, if the local premises are true andAis consistent, thenB is bound to be true. ThatA is consistent is nowhere stated. The fact that the conclusion line is unmarked indicates that (the members of∆∪Θas well as)A∧¬Acan be held to be false, which means thatAis consistent and in that case B is bound to be true if the local premises are true.29 The reasoning is similar for the application of Modus Tollens, except that here the consistency of B matters.
The situation is analogous for all applications of the generic conditional rule RC in CLuNsm and similar inconsistency-adaptive logics.
There is a second reason why the dialetheist should not eschew adaptive logics. The typical abnormality as defined in Section 4. works not only with classical negation, but works equally well with a paraconsistent negation. So where the symbol¬ˇ (defining∗) in (4) is a negation that is paraconsistent and not also paracomplete, (4) still works fine: if all members of the condition can be held to be false, the conclusion follows from the premises.
This comment does not concern (5), the basic rule for negation. This rule, or rather both of them, are still unacceptable for the dialetheist because in it the symbol ¬ˇ in the conclusion needs to be classical. As explained, however, there is no need for¬ˇ to occur anywhere in inconsistency-adaptive logics. Yet there still is a catch. Suppose that the dialetheist position gets generally recognized, that the methodology of the sciences is spelled out in terms of, say, the LP- negation [Priest, 1987], and that scientists would actually apply LP rather than requiring, presupposing and sometimes pretending that their theories be consistent, then the dialetheist might phrase the whole scientific methodology in terms of adaptive logics based on LP. If the condition is false, the local conclusion will follow from the local premises. A hindrance for dialetheists will be that, in the preceding sentence, “false” needs to have the meaning with which I use it: consistently false, not false as meant by Graham Priest [Priest, 1987].
The latter meaning is that A is false iff ¬A is true; this entails that A and
¬A are both false in case they are both true, as may happen in paraconsistent contexts. The situation seems rather crucial. All instances of A∧ ¬A and all instances of A ∧ ¬A ∧¬Bˇ are false in the sense of Priest. There is no point in asking whether they can be held to be false in view of the premises.
They are false in Priest’s sense, now, yesterday, tomorrow andalways because their negation is true, even logically true: ¬(A∧ ¬A) is an LP-theorem; it is LP-equivalent to ¬A∨A. From here on I return to my use of false.
Just for the record, a comment on two related negation-like entities. A para- complete negation, according to whichA and its negation may be jointly false,
29Obviously, from¬A∨B andAfollowsB on the conditionA∧ ¬A.
is insufficient for adaptive logics to work decently. If the ¬ˇ in (4) is paracom- plete, the falsehood of the condition is insufficient for the local conclusion to follow from the local premises. The second negation-like entity is the arrow- bottom construction,A→ ⊥, in which →is a detachable implication and ⊥a bottom operator.30 This is obviously a kind of negation ofA. Dialetheists have argued that A→ ⊥, for →a relevant implication,31 allows them, just as much as the classical logician or intuitionist, to commit themselves to the falsehood of a certain statement A in thatA→ ⊥connects the truth ofA to triviality.
Some paragraphs ago, I argued that there is a problem for dialetheists to apply adaptive logics. Quite unexpectedly, however, there seems to be a way out. I am not a dialetheist, recently I even got doubts on the viability of dialetheism. Yet, these doubts are not related to what follows. Consider the following rules and their typical abnormalities:
¬A ∆
A→ ⊥ ∆∪ {¬A∧ ¬(A→ ⊥)}
A ∆
¬A→ ⊥ ∆∪ {A∧ ¬(¬A→ ⊥)} (6) Dialetheists claim that true inconsistencies are exceptional. So, in non- exceptional situations, that¬Ais given justifies one to defeasibly connect Ato triviality and thatAis given justifies one to defeasibly connect¬Ato triviality.
The typical abnormality may look problematic, but it is not. For most relevant implications, ¬(A→ B) is derivable from A∧ ¬B. Where this is the case,A∧ ¬Ais sufficient to derive both¬A∧ ¬(A→ ⊥) and A∧ ¬(¬A→ ⊥) because¬⊥is a theorem ofLP. To prevent readers from getting overoptimistic, let me point out that the ‘negation’ , defined by A =df A → ⊥, is a paracomplete negation. Clearly, A∨ A is not a theorem unless the relevant
→ is downgraded to a detachable material implication.
It seems to me that the rules and abnormalities in (6) look extremely in- teresting from a dialetheist point of view. They allow dialetheists to express their commitment to the falsehood of a statement in the sense that the false- hood of A connects the truth of A to triviality. Moreover, they may do so without ever using classical negation — dialetheist may continue to catalogue that as a tonk-like operator. So (6) seems to provide a means for dialetheists to apply an inconsistency-adaptive logic without committing themselves to clas- sical negation. Exploring the consequences of this insight obviously deserves a careful study, but that goes beyond the present paper. Moreover, adaptive
30A bottom operator is characterized by the rule “from⊥to deriveA”.
31The intended relevant implications are not those from the well-known and very rich systems devised by Ackermann [Ackermann, 1956], Church [Church, 1951] and especially Anderson and Belnap [Anderson, Belnap 1975; Anderson et al., 1992] but of much weaker systems surveyed by Routley [Routley, 1982] and Brady [Brady, 2006].
consequences derived on a condition ∆, remain to be justified in terms of the joint falsehood, in the dialetheist sense, of the members of∆.
6. In Conclusion
The problem I set out to solve concerned cases where one has an idea for devising an adaptive logic in terms of a defeasible rule. The easier case was where the set of abnormalities was given together with the lower limit logic for ampliative adaptive logics or together with the upper limit logic for corrective adaptive logics. If the idea for the adaptive logic comes in terms of a rule, the set of abnormalities has to be delineated. I presented an extremely simple and transparent recipe for doing so and argued that, except for very special cases, the recipe leads to an adequate result.
A further important point deserves to be mentioned. I have shown that there is a number of formerly unsolved difficulties for dialetheists who try to invoke inconsistency-adaptive logics. For me logics are instruments. Instru- ments may be independent of the philosophical and ideological views of those who use them. So it seems an important feature that adaptive logics as well as the proposed recipe work fine for dialetheists. Disagreements with dialetheists is not an excuse for hiding that, unlike what one might expect, adaptive logics turn out sensible and useful instruments for them.
A very different conclusion is not about generality but about specificity.
There is a huge number of different adaptive logics of inductive generalization.
This is not only required because of the many disagreements between philo- sophers of science on inductive methods. It is also necessary in view of the very different domains of application. To give just one example, the non-logical terms of one language may be well entrenched technical terms and those of another language may be taken straight from natural language. Further dif- ferences will depend on the underlying conceptual framework, on the presence of articulated observational criteria, and so on. Such differences may have an effect on the suitability of a specific inductive method. The situation for other ampliative adaptive logics is analogous.
Similar comments apply to corrective adaptive logics. The upper limit lo- gic is known beforehand, but there are many ways to approach it: different strategies, different lower limit logics, and for each combination of a strategy and lower limit logic, different sets of abnormalities. Of course, not every spe- cific circumstance determines a single adaptive logic. Nevertheless, the choice of a suitable adaptive logic will be largely determined by properties of the the- ory or domain to which it is applied. Mathematical theories have generally conceptual structures that are much simpler that most empirical theories. So they usually require a stronger lower limit logic, validating full Replacement
of Identicals and reducing all statements to truth-functions of literals — truth- functions in the broad sense including quantifiers. But apart from such rough classifications, both mathematical and empirical theories will require careful analysis in order to select the specific non-logical axioms in view of the lower limit logic. Adaptive mathematical theories [Batens, 2014; Batens, 2019] are a case in point.
Part of the importance of the present paper and of the recipe is related to insights that have grown over the years. In the early days, adaptive logics seemed to present an attractive approach to handle certain problems. Examples were (i) inconsistencies coming up unexpectedly in a theory that was intended as consistent or (ii) devising a precise formulation of a given method. By and large, the impression was that adaptive logics were very general tools that could be efficiently applied in nearly all circumstances. Only over the years did it become clear that especially the choice of corrective adaptive logics depends heavily on the context. When a problem is located, adaptive logics do not provide one with a tool that in itself warrants success. One has to carefully choose a language in which to formulate the problem. One has to carefully select the way in which the theory or the data, in which the problem arises, are phrased. Recently, especially with the application to Fregean set theories (sic), it turned out that sometimes one even has to tailor the adaptive logic in view of its application. On the one hand, this shows to what extend Dudley Shapere was right in propagating content guidance and learning how to learn [Shapere, 2004]. On the other hand, it made necessary the search for the present recipe:
content guidance provokes more frequently the need for adaptive logics devised in view of defeasible rules.
Acknowledgements. I want to thank the referee and the editor for their careful work.
References
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